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Mirrors > Home > MPE Home > Th. List > mircgrextend | Structured version Visualization version GIF version |
Description: Link congruence over a pair of mirror points. cf tgcgrextend 25973. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
mirtrcgr.e | ⊢ ∼ = (cgrG‘𝐺) |
mirtrcgr.m | ⊢ 𝑀 = (𝑆‘𝐵) |
mirtrcgr.n | ⊢ 𝑁 = (𝑆‘𝑌) |
mirtrcgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirtrcgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mirtrcgr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
mirtrcgr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
mircgrextend.1 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) |
Ref | Expression |
---|---|
mircgrextend | ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | mirtrcgr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | mirtrcgr.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
9 | mirtrcgr.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐵) | |
10 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircl 26149 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
11 | mirtrcgr.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
12 | mirtrcgr.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
13 | mirtrcgr.n | . . 3 ⊢ 𝑁 = (𝑆‘𝑌) | |
14 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircl 26149 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃) |
15 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mirbtwn 26146 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴)) |
16 | 1, 2, 3, 4, 10, 6, 5, 15 | tgbtwncom 25976 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴))) |
17 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mirbtwn 26146 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋)) |
18 | 1, 2, 3, 4, 14, 12, 11, 17 | tgbtwncom 25976 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋))) |
19 | mircgrextend.1 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) | |
20 | 1, 2, 3, 4, 5, 6, 11, 12, 19 | tgcgrcomlr 25968 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋)) |
21 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircgr 26145 | . . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴)) |
22 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircgr 26145 | . . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋)) |
23 | 20, 21, 22 | 3eqtr4d 2824 | . 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋))) |
24 | 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23 | tgcgrextend 25973 | 1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 distcds 16430 TarskiGcstrkg 25918 Itvcitv 25924 LineGclng 25925 cgrGccgrg 25998 pInvGcmir 26140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-trkgc 25936 df-trkgb 25937 df-trkgcb 25938 df-trkg 25941 df-mir 26141 |
This theorem is referenced by: mirtrcgr 26171 |
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